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Theorem bldisj 15040

Description: Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)

**Assertion**
Ref	Expression
bldisj	⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) = ∅)

Proof of Theorem bldisj Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr3 1010	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))
2		simpr1 1008	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑅 ∈ ℝ^*)
3		simpr2 1009	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑆 ∈ ℝ^*)
4	2, 3	xaddcld 10048	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
5		xmetcl 14991	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ^*)
6	5	adantr 276	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑃𝐷𝑄) ∈ ℝ^*)
7		xrlenlt 8179	. . . . 5 ⊢ (((𝑅 +_𝑒 𝑆) ∈ ℝ^* ∧ (𝑃𝐷𝑄) ∈ ℝ^*) → ((𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄) ↔ ¬ (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
8	4, 6, 7	syl2anc 411	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄) ↔ ¬ (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
9	1, 8	mpbid 147	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ¬ (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆))
10		elin 3367	. . . 4 ⊢ (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) ↔ (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)))
11		simpl1 1005	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝐷 ∈ (∞Met‘𝑋))
12		simpl2 1006	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑃 ∈ 𝑋)
13		elbl 15030	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
14	11, 12, 2, 13	syl3anc 1252	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅)))
15		simpl3 1007	. . . . . . . 8 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → 𝑄 ∈ 𝑋)
16		elbl 15030	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑆 ∈ ℝ^*) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
17	11, 15, 3, 16	syl3anc 1252	. . . . . . 7 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑥 ∈ (𝑄(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
18	14, 17	anbi12d 473	. . . . . 6 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)) ↔ ((𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆))))
19		anandi 592	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ ((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆)) ↔ ((𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) ∧ (𝑥 ∈ 𝑋 ∧ (𝑄𝐷𝑥) < 𝑆)))
20	18, 19	bitr4di 198	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)) ↔ (𝑥 ∈ 𝑋 ∧ ((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆))))
21	11	adantr 276	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑋))
22	12	adantr 276	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑃 ∈ 𝑋)
23		simpr 110	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
24		xmetcl 14991	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
25	21, 22, 23, 24	syl3anc 1252	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑥) ∈ ℝ^*)
26	15	adantr 276	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑄 ∈ 𝑋)
27		xmetcl 14991	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
28	21, 26, 23, 27	syl3anc 1252	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑄𝐷𝑥) ∈ ℝ^*)
29	2	adantr 276	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ ℝ^*)
30	3	adantr 276	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ ℝ^*)
31		xlt2add 10044	. . . . . . . 8 ⊢ ((((𝑃𝐷𝑥) ∈ ℝ^* ∧ (𝑄𝐷𝑥) ∈ ℝ^) ∧ (𝑅 ∈ ℝ^ ∧ 𝑆 ∈ ℝ^*)) → (((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆) → ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆)))
32	25, 28, 29, 30, 31	syl22anc 1253	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆) → ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆)))
33		xmettri3 15013	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)))
34	21, 22, 26, 23, 33	syl13anc 1254	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)))
35	6	adantr 276	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑃𝐷𝑄) ∈ ℝ^*)
36	25, 28	xaddcld 10048	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) ∈ ℝ^*)
37	4	adantr 276	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (𝑅 +_𝑒 𝑆) ∈ ℝ^*)
38		xrlelttr 9970	. . . . . . . . 9 ⊢ (((𝑃𝐷𝑄) ∈ ℝ^* ∧ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ∈ ℝ^*) → (((𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) ∧ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
39	35, 36, 37, 38	syl3anc 1252	. . . . . . . 8 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (((𝑃𝐷𝑄) ≤ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) ∧ ((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
40	34, 39	mpand 429	. . . . . . 7 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (((𝑃𝐷𝑥) +_𝑒 (𝑄𝐷𝑥)) < (𝑅 +_𝑒 𝑆) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
41	32, 40	syld 45	. . . . . 6 ⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) ∧ 𝑥 ∈ 𝑋) → (((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
42	41	expimpd 363	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑥 ∈ 𝑋 ∧ ((𝑃𝐷𝑥) < 𝑅 ∧ (𝑄𝐷𝑥) < 𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
43	20, 42	sylbid 150	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑥 ∈ (𝑃(ball‘𝐷)𝑅) ∧ 𝑥 ∈ (𝑄(ball‘𝐷)𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
44	10, 43	biimtrid 152	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → (𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) → (𝑃𝐷𝑄) < (𝑅 +_𝑒 𝑆)))
45	9, 44	mtod 667	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ¬ 𝑥 ∈ ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)))
46	45	eq0rdv 3516	1 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (𝑅 +_𝑒 𝑆) ≤ (𝑃𝐷𝑄))) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑄(ball‘𝐷)𝑆)) = ∅)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 983 = wceq 1375 ∈ wcel 2180 ∩ cin 3176 ∅c0 3471 class class class wbr 4062 ‘cfv 5294 (class class class)co 5974 ℝ^*cxr 8148 < clt 8149 ≤ cle 8150 +_𝑒 cxad 9934 ∞Metcxmet 14465 ballcbl 14467

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-sep 4181 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-addcom 8067 ax-addass 8069 ax-i2m1 8072 ax-0id 8075 ax-rnegex 8076 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-id 4361 df-po 4364 df-iso 4365 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-fv 5302 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-map 6767 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-xadd 9937 df-psmet 14472 df-xmet 14473 df-bl 14475

This theorem is referenced by: bl2in 15042

W3C validator